Salt effect on the liquid-liquid phase separation of charged macromolecules

Aggregation behaviors of charged macromolecules are ubiquitous in living cells, which can proceed via liquid-liquid phase separation (LLPS). Salt ions play an important role in controlling the LLPS, for which some abnormal phenomena beyond the mean-field Debye-Hϋckel framework have been increasingly observed such as nonmonotonic salt-concentration effect and specific ion effect. In this work, we developed a theory to study the salt effect on the LLPS of charged macromolecules. In our theory, the localized fluctuation of solutes in the dilute phase is considered and the electrostatic fluctuation is also captured by the self-energy of ions. We find that the solubility shows salting-out behavior at low salt concentrations. At high salt concentrations, it can be either salting-in or salting-out, which depends on the dielectric constant of macromolecules. In addition, the solubility also strongly depends on the chemical identity of ions: it follows the inverse Hofmeister series at low salt concentrations whereas it shows direct Hofmeister series at high salt concentrations.  These results are in agreement with experimental observations. Furthermore, we analytically obtained a universal curve determining the borderline between salting-in and salting-out regions, which matches quantitatively with the results of a variety of charged macromolecule systems such as lysozyme, elastin-like polypeptide and PNIPAM in sodium halogen solutions.